Coherent sheaves on Finite Sets
Coherent sheaves on a finite set $X$ are characterized by the stalks of the the elements $x \in X$. I.e. a coherent sheaf on $X$ is a collection of vector spaces $V_x$ for each $x\in X$. More generally for a group $G$ and a G-set $X$ we can define consider $G$-equivariant sheaves on $X$. In this case a sheaf is specified by representations of the stabilizers of representatives for the orbits.
Equivariant Coherent Sheaves
We provide the datatype
CoherentSheafObject <: ObjectThe category of equivariant coherent sheaves has type
CohSheaves <: MultiTensorCategoryand can be constructed via
TensorCategories.coherent_sheaves — Function
Morphisms are given by morphisms of representations of the stalks and are of type
CohSheafMorphism{T,G} <: MorphismAbstractAlgebra.compose — Method
AbstractAlgebra.direct_sum — Method
direct_sum(f::CohSheafMorphism, g::CohSheafMorphism)Return the direct sum of morphisms of sheaves.
sourceAbstractAlgebra.direct_sum — Method
direct_sum(X::CohSheafObject, Y::CohSheafObject)Return the direct sum of sheaves. Return also the inclusion and projection.
sourceAbstractAlgebra.is_isomorphic — Method
is_isomorphic(X::CohSheafObject, Y::CohSheafObject)Check whether $X$and $Y$ are isomorphic and the isomorphism if possible.
sourceAbstractAlgebra.kernel — Method
kernel(f::CohSheafMorphism)Return a tuple (K,k) where K is the kernel object and k is the inclusion.
Hecke.cokernel — Method
cokernel(f::CohSheafMorphism)Return a tuple (C,c) where C is the kernel object and c is the projection.
Hecke.decompose — Method
decompose(X::CohSheafObject)Decompose $X$ into a direct sum of simple objects with multiplicity.
sourceHecke.dual — Method
Hecke.is_semisimple — Method
Hecke.tensor_product — Method
tensor_product(f::CohSheafMorphism, g::CohSheafMorphism)Return the tensor product of morphisms of equivariant coherent sheaves.
sourceHecke.tensor_product — Method
tensor_product(X::CohSheafObject, Y::CohSheafObject)Return the tensor product of equivariant coherent sheaves.
sourceTensorCategories.Hom — Method
TensorCategories.Pullback — Method
Pullback(C::CohSheaves, D::CohSheaves, f::Function)Return the pullback functor C → D defined by the G-set map f::X → Y.
TensorCategories.Pushforward — Method
Pushforward(C::CohSheaves, D::CohSheaves, f::Function)Return the push forward functor C → D defined by the G-set map f::X → Y.
TensorCategories.associator — Method
associator(X::CohSheafObject, Y::CohSheafObject, Z::CohSheafObject)Return the associator isomorphism (X⊗Y)⊗Z → X⊗(Y⊗Z).
TensorCategories.braiding — Method
TensorCategories.coev — Method
TensorCategories.coherent_sheaves — Method
TensorCategories.coherent_sheaves — Method
TensorCategories.ev — Method
TensorCategories.simples — Method
TensorCategories.spherical — Method
TensorCategories.stalks — Method
TensorCategories.zero_morphism — Method
Convolution Category
The objects of this category are again $G$-equivariant coherent sheaves on a finite $G$-set $X\times X$. But we endow them with a different monoidal product.
Let $p_{ij}: X\times X\times X \to X \times X$ be the canonical projections. Then we define the monoidal product of two coherent sheaves $x$ and $y$
\[\begin{aligned} x\otimes y = p_{13}_\ast(p_{12}^\ast(x)\otimes' p_{23}^\ast(y)) \end{aligned}\]
where $\otimes'$is the monoidal product of $Coh(X\times X\times X)$. Similar for morphisms.
Objects in this category are of type
ConvolutionObject <: Objectwhile the convolution category is of type
ConvolutionCategory <: MultiTensorCategoryand can be constructed by
TensorCategories.convolution_category — Function
convolution_category( K::Field, X::GSet)Return the category of equivariant coherent sheaves on $X$ with convolution product.
sourceconvolution_category(K::Field, X)Return the category of coherent sheaves on $X$ with convolution product.
sourceMorphisms are just morphisms of coherent sheaves with the new tensor product.
ConvolutionMorphism <: MorphismAbstractAlgebra.direct_sum — Method
direct_sum(f::ConvolutionMorphism, g::ConvolutionMorphism)Return the direct sum of morphisms of coherent sheaves (with convolution product).
sourceAbstractAlgebra.direct_sum — Method
AbstractAlgebra.is_isomorphic — Method
is_isomorphic(X::ConvolutionObject, Y::ConvolutionObject)Check whether $X$and $Y$are isomorphic. Return the isomorphism if true.
sourceHecke.decompose — Method
decompose(X::ConvolutionObject)Decompose $X$ into a direct sum of simple objects with multiplicities.
sourceHecke.is_semisimple — Method
Hecke.tensor_product — Method
tensor_product(f::ConvolutionMorphism, g::ConvolutionMorphism)Return the convolution product of morphisms of coherent sheaves.
sourceHecke.tensor_product — Method
tensor_product(X::ConvolutionObject, Y::ConvolutionObject)Return the convolution product of coherent sheaves.
sourceTensorCategories.Hom — Method
TensorCategories.convolution_category — Method
convolution_category(K::Field, X)Return the category of coherent sheaves on $X$ with convolution product.
sourceTensorCategories.convolution_category — Method
convolution_category( K::Field, X::GSet)Return the category of equivariant coherent sheaves on $X$ with convolution product.
sourceTensorCategories.orbit_stabilizers — Method
orbit_stabilizers(C::ConvolutionCategory)Return the stabilizers of representatives of the orbits.
source